Triangles and regular polygons: when equal sides let you multiply

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What to assume they know

Perimeter as the distance all the way round: perimeter as the sum of a polygon's side lengths
The rectangle and square formulas are shortcuts for the same addition: the square rule, and why equal sides turn a sum into a product
Shapes come in sequences too, with rules of their own: Shape Sequence #1 in Chapter 1, the sequence of regular polygons this section points back to
Acute, obtuse and reflex: one classification covering every angle: what it means for two angles to be equal

What they should be able to do

Compute the perimeter of a triangle from three given side lengths
Find a missing side of a triangle from its perimeter and the other two
State what the book means by a regular polygon, in both its parts
Derive the equilateral-triangle rule from the general triangle rule
Say what an equilateral triangle and a square have in common that lets both rules exist
Extend the pattern to a five-sided and a six-sided equal-sided figure, and state the general rule in words
Distinguish the property the shortcut needs (equal sides) from the full definition of regular (equal sides and equal angles)
Recognise when a polygon is irregular and the shortcut must not be used

Where it usually goes wrong

"Regular means ordinary or usual." It is a technical word here with a two-part definition, and students read it as the everyday one. Say both parts aloud every time the term is used in the first half of the explanation.
"Any triangle's perimeter is three times a side." The chapter opens the section with a 4-5-7 triangle for exactly this reason. Multiplying by three requires the sides to be equal, and most triangles' sides are not.
"A rectangle is regular, because its opposite sides are equal." Regular needs all sides equal, not opposite pairs. A non-square rectangle has equal angles and unequal sides, so it fails the first half of the definition while passing the second — a useful case to show, since it separates the two halves.
"Equal sides is the same thing as regular." It is not, and the shortcut only ever uses the sides. A four-sided figure can have four equal sides with corners that are not square; its perimeter is still four times a side, and it is still not what this book calls regular. Flag the shape as an added example — this chapter does not print one.
"Two examples prove the general rule." The book gives three sides and four sides and then stops.
"A longer string makes a bigger side, whatever the shape." Question 5 has one string and three answers; the side length falls as the side count rises.
"Perimeter of a hexagon = 6 × side." Only if the six sides are equal. The book's question says so explicitly.

Questions to check understanding

Perimeter of a triangle from three sides; a missing side from the perimeter
Perimeter of an equilateral triangle, a regular pentagon, a regular hexagon
Divide a fixed length into a stated number of equal sides
Decide whether a described polygon is regular, and justify with both conditions
State the general rule for a regular polygon in words
Identify from a drawing which figures the multiplication shortcut applies to
The answer key appended to the cached PDF answers question 5 of the p.132 set on its footer page 1, and gives the general regular-polygon rule on its footer page 3 in answer to the open task of p.136

Examples worth working on the board

The scalene case (p.131). A triangle drawn with 5 cm and 4 cm sloping up to an apex and 7 cm along the base. The book adds the three and prints 16 cm. Nothing repeats, so nothing collapses — start the explanation here so the shortcut is later seen as a special mercy rather than the normal state of affairs.
The reverse question (p.132, question 3). A triangle of perimeter 55 cm with two sides of 20 cm and 14 cm; the third is wanted.
The equilateral triangle (p.135). Drawn as triangle ABC with no measurements marked. The book writes the perimeter as AB + BC + AC, replaces BC and AC by AB because they are equal, and reaches three times one side. This is the same collapse the square got on p.130.
The definition of regular (p.135). Every side the same length and every angle the same size. The book names the equilateral triangle and the regular pentagon as its examples and spells the two conditions out inside each name, three of each for the triangle and five of each for the pentagon. It also points the reader back to Shape Sequence #1 in Chapter 1.
The book's closing question (p.135). What an equilateral triangle has in common with a square. Leave it open for a beat before answering; it is the pivot of the whole topic.
The string, question 5 on p.132. One 36 cm string is bent in turn into three equal-sided figures — four sides, then three, then six. The book supplies its own gloss for the six-sided one inside the question. Three divisions of a single length.
The open task and the Teacher's Note (p.136). The reader is asked to find regular-shaped objects around them, measure their perimeters, and generalise the rule to other regular polygons; the Teacher's Note asks teachers to push students towards a general formula. The chapter itself never prints one — the general statement appears only in the answer key appended to the cached PDF, where it is given as the number of sides multiplied by the length of a side.
Values worth carrying into the figure. A regular pentagon of side 5 cm has perimeter 25 cm; a regular hexagon of side 5 cm has perimeter 30 cm; the square and equilateral triangle of side 5 cm give 20 cm and 15 cm. These four are an added worked instances, not the book's, and are there to let the pattern be seen as the side count rises.

Figures to have open

The 4-5-7 triangle as printed on p.131, with the three lengths marked. Standard schematic.
Triangle ABC lettered as on p.135, unmeasured, so the letters can be substituted.
A row of regular polygons — equilateral triangle, square, regular pentagon, regular hexagon — all drawn with the same side length, so the perimeters can be compared by eye as well as by arithmetic. Standard schematic; the book draws no such row in this chapter, though Chapter 1's Shape Sequence #1 is its ancestor.
One equal-sided four-sided figure that is not a square, for section 10. An added figure; not printed in this chapter.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 6 "Perimeter and Area", §6.1 "Perimeter", p.131 (the sub-heading "Perimeter of a triangle"), p.132 (questions 3 and 5), p.135 (the sub-headings "Perimeter of a regular polygon" and "Perimeter of an equilateral triangle", and the closing question) and p.136 (the open task and the Teacher's Note)
Chapter 1, Shape Sequence #1, referred to by name on p.135 as the place regular polygons were first met — see Shapes come in sequences too, with rules of their own
Chapter Summary, p.150, which states the polygon rule but prints no regular-polygon rule
Companion topic The rectangle and square formulas are shortcuts for the same addition for the square and rectangle derivations this one parallels
Solutions appendix bound into the cached PDF after the book's last printed page, footer pages 1 and 3